The Geometry of F4-Models Mboyo Esole♠, Patrick Jefferson♣, Monica Jinwoo Kang♣ ♠ Department of Mathematics, Northeastern University 360 Huttington Avenue, Boston, MA 02115, USA ♣ Department of Physics, Jefferson Physical Laboratory, Harvard University 17 Oxford Street, Cambridge, MA 02138, U.S.A [email protected], [email protected], [email protected] Abstract: We study the geometry of elliptic fibrations satisfying the conditions of Step 8 of Tate’s algorithm. We call such geometries F4-models, as the dual graph of their special fiber is the twisted affine Dynkin t diagram F4. These geometries are used in string theory to model gauge theories with the exceptional Lie group F4 on a smooth divisor S of the base. Starting with a singular Weierstrass model of an F4-model,e we present a crepant resolution of its singularities. We study the fiber structure of this smooth elliptic fibration and identify the fibral divisors up to isomorphism as schemes over S. These are P1-bundles over S or double covers of P1-bundles over S. We compute basic topological invariants such as the double and triple intersection numbers of the fibral divisors and the Euler characteristic arXiv:1704.08251v1 [hep-th] 26 Apr 2017 of the F4-model. In the case of Calabi-Yau threefolds, we compute the linear form induced by the second Chern class and the Hodge numbers. We also explore the meaning of these geometries for the physics of gauge theories in five and six-dimensional minimal supergravity theories with eight supercharges. We also introduce the notion of “frozen representations” and explore the role of the Stein factorization in the study of fibral divisors of elliptic fibrations. Keywords: Elliptic fibrations, Crepant morphism, Resolution of singularities, Weierstrass models Contents 1 Introduction 2 1.1 G-modelsandF-theory................................. 2 1.2 F4-models: definition and first properties . ........ 3 1.3 Representations associated to an F4-modelandflops. ................. 4 1.4 Counting hypermultiplets: Witten’s genus formula . ............... 5 1.5 Arithmetic versus geometric degenerations . .............. 6 1.6 Frozen representations . ........ 7 1.7 Summaryofresults................................ ..... 8 2 Basic conventions and definitions 9 2.1 Geometricweights ................................ ..... 9 2.2 Step 8 of Tate’s algorithm . ....... 10 3 Crepant resolution 11 4 Fiber structure 13 4.1 Structure of the generic fiber . ........ 14 4.2 Fibraldivisors.................................. ...... 14 4.3 Representation associated to the elliptic fibration . ................. 18 5 Topological invariants 19 5.1 Stein factorization and the geometry of non-simply laced G-models . 21 6 ApplicationtoM-theoryandF-theoryin5and6dimensions 23 6.1 Intriligator-Morrison-Seiberg potential . ................ 23 6.2 Six-dimensional uplift and anomaly cancellation conditions............... 24 6.3 Cancellations of six-dimensional anomalies for an F4-model .............. 25 6.4 Frozen representations . ........ 26 6.5 Geometry of fibral divisors in the case of frozen representations . 28 7 Conclusion 29 A Weierstrass models and Deligne’s formulaire 33 B Representation theory of F4 34 C A double cover of a ruled surface branch along 2b fibers 34 1 1 Introduction An elliptic fibration is a proper projective morphism ϕ : Y −→ B between normal varieties such that the generic fiber is a nonsingular projective curve of genus one and the fibration is endowed with a rational section. Under mild assumptions, an elliptic fibration is birational to a potentially singular Weierstrass model [23,65]. The locus of points of B over which the elliptic fiber is singular is called the discriminant locus. The discriminant locus of a Weierstrass model is a Cartier divisor that we denote by ∆. The type of the fiber over the generic point of an irreducible component of the discriminant locus of an elliptic fibration is well understood following the work of Kodaira [53], Néron [66], and Tate [70]. Fibers over higher dimensional loci are not classified and are the subject of much interdisciplinary research by both mathematicians and physicists [3,10,21,27,28,35–38,40– 42, 54–56, 58, 60, 61, 68, 71]. Crepant resolutions of a Weierstrass model are relative minimal models in the sense of Mori’s program [59]. Different crepant resolutions of the same Weierstrass model are connected to each other by a sequence of flops. 1.1 G-models and F-theory A classical problem in the study of elliptic fibrations is understanding the geometry of the crepant resolutions of Weierstrass models and their flop transitions. A natural set of singular Weierstrass models to start with are the G-models. The constructions of G-models are deeply connected to the classification of singular fibers of Weierstrass models and provide an interesting scene to explore higher dimensional elliptic fibrations with a view inspired by their applications to physics. We follow the definitions and notation of Appendix C of [32]. The framework of G-models naturally includes a geometric formulation of basic notions of representation theory such as the theory of root systems and weights of representations. The data characterizing a G-model can be understood in the framework of gauge theories, which provides a natural language to talk about the geometry of these elliptic fibrations. F-theory enables a description of gauge theories in string theory and M-theory via geometric engineering based on elliptic fibrations [12,63,64,72]. The data of a gauge theory that can be extracted from an elliptic fibration are its Lie algebra, its Lie group, and the set of irreducible representations defining how charged particles transform under the action of the gauge group. In F-theory, the Lie algebra is determined by the dual graphs of the fibers over the generic points of the irreducible components of the discriminant locus of the elliptic fibration. The Mordell-Weil group of the elliptic fibration is conjectured to be isomorphic to the first homotopy group of the gauge group [22]. Hence, the Lie group depends on both the singular fibers and the Mordell-Weil group of the elliptic fibration. In F-theory, the singular fibers responsible for non-simply laced Lie algebras are not affine Dynkin diagrams, but twisted affine Dynkin diagrams, as presented in Table 1 and Figure 1.1, respectively, on pages 4 and 32. These twisted affine Dynkin diagrams are the Dynkin duals of the corresponding affine Dynkin diagrams. The Dynkin dual of a Dynkin diagram is obtained by inverting all the arrows. In the language of Cartan matrices, two Dynkin diagrams are dual to each other if their Cartan matrices are transposes of each other. The Langlands duality interchanges Bn and Cn, but preserves all the other simple Lie algebras. For affine Dynkin diagrams, only the ADE series are preserved under the Langlands duality. In particular, the Langlands duals t t t t of Bn, Cn, G2 and F4 are respectively denoted in the notation of Carter as Bn, Cn, G2 and F4. e e e e e e e e 2 1.2 F4-models: definition and first properties One of the major achievements of F-theory is the geometric engineering of exceptional Lie groups. These elliptic fibrations play an essential role in the study of superconformal field theories even in absence of a Lagrangian description. The study of non-simply laced Lie algebras in F-theory started in May of 1996 during the second string revolution with a paper of Aspinwall and Gross [5], followed shortly afterwards by the classic F-theory paper of Bershadsky, Intriligator, Kachru, Morrison, Sadov, and Vafa [12]. M-theory compactifications giving rise to non-simply laced gauge groups are studied in [25, 50]. In this paper, we study the geometry of F4-models, namely, G-models with G = F4, the excep- tional simple Lie group of rank 4 and dimension 52. F4 is a simply connected and non-simply laced Lie group. An F4-model is mathematically constructed as follows. Let B be a smooth projective variety of dimension two or higher. Let S be an effective Cartier divisor in B defined as the zero scheme of a section s of a line bundle S . Since S is smooth, the residue field of its generic point is a discrete valuation ring. We denote the valuation with respect to S as vS. AnF4-model is defined by the crepant resolution of the following Weierstrass model 2 3 3+α 2 4 3 y z = x + s a4,3+αxz + s a6,4z , α ∈ Z≥0, (1.1) where vS(a6,4) = 0, and either vS(a4,3+α) = 0 or a4,3+α = 0. We assume that a6,4 is generic; in ∗ns particular, a6,4 is not a perfect square modulo s. This ensures that the generic fiber is of type IV rather than IV∗s. The defining equation (1.1) is of type (c6) in Néron’s classification of minimal Weierstrass mod- els, defined over a perfect residue field of characteristic different from 2 and 3 [66]. This corresponds to Step 8 of Tate’s algorithm [70]. Hence, the geometric fiber over the generic point of S is of Kodaira type IV∗. As discussed earlier, the generic fiber over S has a dual graph that is the twisted t Dynkin diagram F4, which is the Langlands dual of the affine Dynkin diagram F4.The structure of the generic fiber of the F4-model is due to the arithmetic restriction that a6,4 is not a perfect square modulo s. e e The innocent arithmetic condition which characterizes a F4-model has surprisingly deep topo- logical implications for the elliptic fibration. When a6,4 is a perfect square modulo s, the elliptic ∗ns ∗s fibration is called an E6-model; the generic fiber IV is replaced by the fiber IV whose dual graph is the affine Dynkin diagram E6.